deletions | additions       diff --git a/Magnetic response.tex b/Magnetic response.tex  index 23316b8..f7b75ab 100644  --- a/Magnetic response.tex  +++ b/Magnetic response.tex 
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%  \begin{equation}  \label{eq:mag1}  \delta\hat{v}^{\mathrm{mag}} \delta\hat{\vec{v}}^{\mathrm{mag}}  = \frac1{2c}({\vec{r}}\times{\vec{p}})\cdot\vec{B} \frac1{2c}({\vec{r}}\times{\vec{p}})  = \frac1{2c}\vec{L}\cdot\vec{B} \frac1{2c}\vec{L}  \end{equation}  %  and  %  \begin{equation}  \label{eq:mag2}  \delta^2{V}^{\mathrm{mag}} \delta^2\hat{v}^{\mathrm{mag}}  = \frac1{8c^2}(\vec{B}\times{\vec{r}})^2\ \frac1{8c^2}(\vec{\mathrm{B}}\times{\vec{r}})^2\  . \end{equation}  The induced magnetic moment can be expanded in terms of the external magnetic field which, to first order, reads  %  \begin{equation}  \label{eq:bchi}  m_i=m^0_i+\sum_i\chi_{ij}B^{ext}_j\ m_i=m^0_i+\sum_j\chi_{ij}\mathrm{B}^{ext}_j\  , \end{equation}  %  where \(\vec{\chi}\) is the magnetic susceptibility tensor. The For finite systems the  permanent magnetic moment can be calculated directly from the  ground-state wave-functions as  %  \begin{equation}  \label{eq:magneticmoment}  \vec{m}^0=  \sum_k\langle\varphi_k|\delta{\vec{V}}^{\mathrm{mag}}|\varphi_{k}\rangle\ \sum_n\langle\varphi_n|\delta{\hat{\vec{V}}}^{\mathrm{mag}}|\varphi_{n}\rangle\  . \end{equation}  %  For the susceptibility, we need to calculate the first order response